4:["$","div",null,{"className":"min-h-screen bg-white flex flex-col","children":[["$","div",null,{"className":"px-6 pt-6 pb-4 border-b border-gray-100","children":[["$","div",null,{"className":"flex items-center gap-1.5 text-xs text-gray-400 mb-4","children":[["$","a",null,{"href":"/","className":"hover:text-[#26a69a] transition-colors","children":"Market"}],["$","span",null,{"children":"›"}],["$","span",null,{"className":"text-gray-600","children":"Tarski monster group"}]]}],["$","div",null,{"className":"text-[10px] text-gray-400 uppercase tracking-widest mb-1","children":"Company Profile"}],["$","h1",null,{"className":"text-2xl font-bold text-gray-900","children":"Tarski monster group"}],null]}],["$","div",null,{"className":"flex-1 px-6 py-6","children":[["$","p",null,{"className":"text-sm text-gray-600 leading-relaxed mb-8","children":"In the area of modern algebra known as group theory, a Tarski monster group, named for Alfred Tarski, is an infinite group such that every proper subgroup, other than the identity subgroup, is a cyclic group of order a fixed prime number p. A Tarski monster group is necessarily simple. It was shown by Alexander Yu. Olshanskii in 1979 that Tarski groups exist, and that there is a Tarski p-group for every prime p > 1075. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture."}],["$","div",null,{"className":"columns-1 md:columns-2 gap-10","children":[["$","div","0",{"className":"mb-8 break-inside-avoid","children":[["$","div",null,{"className":"text-xs font-semibold text-gray-400 uppercase tracking-wide mb-3 border-b border-gray-100 pb-1","children":"Definition"}],["$","div",null,{"className":"text-sm text-gray-600 leading-relaxed wiki-content","dangerouslySetInnerHTML":{"__html":"A Tarski group is an infinite group such that all proper subgroups have prime power order. Such a group is then a Tarski monster group if there is a prime p such that every non-trivial proper subgroup has order p. An extended Tarski group is a group G that has a normal subgroup N whose quotient group G/N is a Tarski group, and any subgroup H is either contained in or contains N. ==Properties=="}}]]}],["$","div","1",{"className":"mb-8 break-inside-avoid","children":[["$","div",null,{"className":"text-xs font-semibold text-gray-400 uppercase tracking-wide mb-3 border-b border-gray-100 pb-1","children":"Properties"}],["$","div",null,{"className":"text-sm text-gray-600 leading-relaxed wiki-content","dangerouslySetInnerHTML":{"__html":"As every group of prime order is cyclic, every proper subgroup of a Tarski monster group is cyclic. As a consequence, the intersection of any two different proper subgroups of a Tarski monster group must be the trivial group. • Every Tarski monster group is finitely generated. In fact it is generated by every two non-commuting elements. • If G is a Tarski monster group, then G is simple. If N\\trianglelefteq G and U\\leq G is any subgroup distinct from N the subgroup NU would have p^2 elements. • The construction of Olshanskii shows in fact that there are continuum-many non-isomorphic Tarski Monster groups for each prime p>10^{75}. • Tarski monster groups are examples of non-amenable groups not containing any free subgroups. ==References=="}}]]}]]}],"$Lf"]}],"$L10"]}]